Optimal edge-based shape detection

We propose an approach to accurately detecting two-dimensional (2-D) shapes. The cross section of the shape boundary is modeled as a step function. We first derive a one-dimensional (1-D) optimal step edge operator, which minimizes both the noise power and the mean squared error between the input and the filter output. This operator is found to be the derivative of the double exponential (DODE) function, originally derived by Ben-Arie and Rao (1994). We define an operator for shape detection by extending the DODE filter along the shape's boundary contour. The responses are accumulated at the centroid of the operator to estimate the likelihood of the presence of the given shape. This method of detecting a shape is in fact a natural extension of the task of edge detection at the pixel level to the problem of global contour detection. This simple filtering scheme also provides a tool for a systematic analysis of edge-based shape detection. We investigate how the error is propagated by the shape geometry. We have found that, under general assumptions, the operator is locally linear at the peak of the response. We compute the expected shape of the response and derive some of its statistical properties. This enables us to predict both its localization and detection performance and adjust its parameters according to imaging conditions and given performance specifications. Applications to the problem of vehicle detection in aerial images, human facial feature detection, and contour tracking in video are presented.